26 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA The lower envelope of an infinite family of lower semicontinuous functions is not necessarily lower semicontinuous. As an example, for each rational number r let/r denote the characteristic function of R — {r}, which is lower semicontinuous (12.7.4). The lower envelope / of the denumerable family C/r)reQ takes the value 0 at each rational number and the value 1 at each irrational number (in other words, it is the " Dirichlet function" 1 — <pQ, cf. (3.11.6)), and is not lower semicontinuous at irrational points. The following proposition is a partial converse of (12.7.7): (12.7.8) Let E be a separable, metrizable, locally compact space, and let f be a lower semicontinuous mapping ofE into R. Suppose that there exists a finite real valued function g, continuous on E and with compact support, such that f^.g (which implies that/(.x) ^ 0 in the complement of some compact set). Then there exists an increasing sequence (/,) of'finitereal valued functions, con- tinuous on E and with compact supports, such that f = sup fn. n Replacing/if necessary by/— g (which is everywhere defined), we may assume that/g 0. It is then enough to show that there exists a denumerable family of finite real-valued continuous functions with compact supports, with /as upper envelope. For it we arrange this family in a sequence (gn) and take /„ = sup(#1? ..., gn), it is clear that the/, satisfy the required conditions. Let (Un) be an increasing sequence of relatively compact open sets in E, such that E = (J Un (3.18.3). If gn is the function which agrees with/on UM 71 and is zero on E — Un, then clearly/= sup#n, and gn is lower semicontin- n uous. Hence if gn is the upper envelope of a denumerable set Dn of finite real- valued continuous functions with compact supports, then / will be the upper envelope of (J DM (2.3.6). Hence we may restrict ourselves to the case where n f has compact support. Suppose first of all that/= <pA, where A is a relatively compact open set. Put/M(;c) = n - inf(d(x, E - A), l/ri). This is a finite, continuous, positive- valued function on E, with compact support (3.11.8). Moreover, we have /n(X) =/W whenever x e E — A or d(x, E — A) ^ l/n. It follows immediately that/=sup/n (3.8.9). n In the case where/^ 0 and has compact support, we can restrict ourselves to the case where 0 g/^ 1, provided that the functions/, are chosen so that 0 ^fn(x) < 1 for all x e E, For in the general case the function h =//(! +/) (taken to have the value 1 at points where/(X) = + oo) is lower semicontinu- ous (cf. (3.3.2)). If we have h = sup hn, with hn continuous and hn(x) < I forother words, Supp(/) is the